I heard that it may be possible to install the Java 7 JDK (including a JRE) to regain preferences and enable switching. If that is the only course of action I will do it, but I would much prefer JAVA_HOME to work properly with Java 6 without Java 7.

That appears to work. Still testing other stuff. The page sparkles with references such as "if, after installing Java for OS X 2012-006 and Java 7", and I read of problems with it (here as well), so it didn't look like it applied. But, it does, even if you don't install Java7. Also, try to avoid basically writing a new question when you edit. We see them as our content, not community (right or wrong). SO is less overbearing, but edits there seem by-fiat and based on personal taste.
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Andrew BackerNov 27 '12 at 3:39

Sorry, the intention wasn't to overbear, I tried hard to keep the essence of the question whilst making uppermost the key points etc.. Feel free to revert or even better, make your own edit. You make very good points about Apple's document – I'll request an improvement. Thanks.
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Graham PerrinNov 27 '12 at 21:03

1

This worked for more applets I have tried. In the start I just tried one. The most important (citrix) still has an issue that might be because of the update, but I think that is unrelated.
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Andrew BackerNov 28 '12 at 3:37

No, the OS X 10.8.2 update didn't remove the Java 6 setup on your system - but the Java update 2012-006 where you got the JRE/JDK 6 update 37 did that - support.apple.com/kb/HT5493
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Rene LarsenNov 26 '12 at 14:12

You are right. I don't know what I was thinking, i'm sorry about that. It must have been something I ate. Even my question mentions the update as doing it, not ML. I think I just installed them back-to-back basically. If you edit the question (however briefly) I'll fix my foolish vote.
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Andrew BackerNov 28 '12 at 3:36